Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

332 MR RICIHMOND, ON MINIMAL SURFACES.
coordinate planes about the z-axis will bring both these numerical coefficients to the
same real value A. For real values of p the typical real component fractions are
therefore
A.,k () {(l + im) + (l - i)k} - (1 + ); ( = 2, 3, 4,...)
Every real minimal value of p which is a rational function of 1, m, n, may be expressed
as the sutm of a finite number of real fractions, each separately reducible by real transformation of axes to one of the forms just quoted. Terns such as
al + f3m + yn
may also be present, but are ignored since a change of origin will remove them.
If we again introduce Euler's angles 0 andcl, as in ~ 4, the surface corresponding
to the above value is
p = A. k (n). {(l + i)k + ( - im)k} ( + m2); ( = 2, 3, 4,...)
= B. cos C. {(k- cos 0) (cot 6 - (k + cos 0) t-tan 2 0}
and may be described as the standard minimal multiple paraboloid of the kth type: the
origin of coordinates is called its centre and the z-axis its axis. The class of every real
multiple paraboloid that is a minimal surface is necessarily odd; thus the above standard
surface is a 2k-fold paraboloid and is of class 2k +1. The theorem established now admits
of the following statement:By placing a finite numinber of standard surfaces (defined above) with their centres coinciding but zoith various orientations, and taking the locus of the centre of mean position
of the points of contact of parallel tangent planes, we can obtain every minimal surface
which is a multiple paraboloid.
Corresponding to any selected real direction, a multiple paraboloid has, as was pointed
out, one and only one tangent plane; there is therefore no ambiguity in the foregoing
construction: certain of the planes may however be at an infinite distance. If the surface
be minimal, the number of infinitely distant tangent planes must be finite, their directions
being normal to the axes of the standard surfaces from which the given surface may be
derived. Given a minimal multiple paraboloid, the directions of the axes of the component
surfaces are thus plain geometrically.